7359-1-F Copy____ THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory ACOUSTIC AND ELECTROMAGNETIC SCATTERING PROBLEMS 'FOR LOW -FREQUEICIES FINAL REPORT ON NSF GRANT GP-4581 by Ergun Ar November 1966 NSF Grant No. GP-4581 7359-1-F = RL-2154 Contract With: National Science Foundation Washington, D. C. 20550 Administered through: OFFICE OF RESEARCH ADMINISTRATION * ANN ARBOR

THE UNIVERSITY OF MICHIGAN 7359-1-F ACOUSTIC AND ELEC TRlETIC SCATTERING PROIBLDMS FOR LOW REQU-ENCIES by Ergun Ar -- Final Report on NSF Grant GP-4581 June 1965 - November 1966 November 1966 Prepared for National Science Foundation Washington, D. C. 20550

THE UNIVERSITY OF MICHIGAN 7359-1-F I. Introduction The classic three-dimensional scalar scattering problem consists of determining a function 0 exterior to a smooth finite boundary B, which function is a solution of a scalar Helmholtz equation, satisfies Dirichlet or Neumann boundary conditions on B, and obeys a radiation condition at infinity, i.e., (7+k2) + 0 o 0(1) i or a - - on B, (2) 0n an lim r( a 0s - ikb ) = 0, (3) ar-o r -o ao where 0 is the incident field which is known everywhere including the boundary G. The study of the relation between this problem and potential problems (boundary value problems for the Laplace's equation, V 0 = 0) goes back to 1897. The general problem is one of generating solutions of the Helmholtz equation (vector or scalar), which satisfy prescribed conditions on a given boundary in terms of solutions of Laplace's equation. Physically, this amounts to an attempt to infer the manner in which an obstacle perturbs the field due to a source of wave motion from a knowledge of how the same object perturbs a stationary (non-oscillatory) field, e.g., determining an electromagnetic field from an electrostatic field. The advantage of such a procedure derives from the fact that stationary fields re physically simpler than wave phenomena, and -tbe -aboclated matlematieal problems, Dougtt often still formidable, are always more easily handled. 1

THE UNIVERSITY OF MICHIGAN 7359-1-F Interest in this problem has gained new momentum in recent years. The major drawback in most of the methods heretofore proposedis iit their intrinsic dependence on a particular geometry. That is, the techniques result from the exploitation of the geometric properties of the surface on which the boundary conditions are specified. For those shapes where the Helmholtz equation is separable, of course, the low frequency expansion may always be obtained from the series solution provided sufficient knowledge of the special functions involved is available. Most low frequency techniques, however, have as their starting point the formulation of scattering problems as integral equations using the Helmholtz representation of the solution in terms of its properties on the boundary and the free space Green's function, i.e., 5 ~I 47r t ( PB) a ^,BB - 'B 'an 0 "B'B dB (4) B where ikR(p, p B) e u= R(p, PB) the integration is carried out over the entire scattering surface B, the normal here is taken out of B, p is the general field point, and pB a point on B whose coordinates are the integration variables, and R is the distance between them. This formulation is also vital to the proof of existence of solutions for a general boundary. Noble (7) shows how the integral formulation solutions for a general boundary. Noble shows how the Integral formulation 2

THE UNIVERSITY OF MICHIGAN 7359-1-F (4) may be used to obtain a representation of the solution of a scattering problem for a general bounday as a perturbation of the solution of the corresponding potential problem. Each term in the low frequency expansion is the solution of an integral equation which differs only in its inhomogeneous part from term to term. However, this formulation does not yield an explicit representation for successive terms in general except as the formal Inverse. Long sought has been the development of a systematic procedure which will generate the solution of the Helmholtz equation, satisfying a particular boundary conditions from the solution of Laplace's equation which satisfies the same boundary condition. In this connection Kleinman(5) has made the following key observation. If l(a) V is the volume exterior to a smooth, closed, and bounded surface B, 1(b) 1 G (p,) - p, p u(P, o o 4irR(p, p o o 0 is the potential Green's function of the first kind ( G (PB' p) = 0), and ikR(p,p ) 0 G (p' P) + ue..p k o 47rR(p, p + Uk (' Po) is the Green's function for the Helmholtz equation, also satisfying ariDlhitlutt condition on B, then the scattered field uk (p, p) satisfies the integral equation 3

THE UNIVERSITY OF MICHIGAN 7359-1-F kr G(PI PI 2) f _ uk(PP = P 2ike O | V rle k (P1, po)] dV1 ikr e + e4r Br -ikrB + ikR (pB, p) e a R(Pp P) an B'o G (, pB) da (5) Here dv is a volume element in coordinates P1 = (1, 0' r1) and do is a surface element and a/8n the normal derivative directed out of V expressed in coordinates PB. R(p,p ) is the distance between the points p and p. The origin of the spherical coordinates p = (r, 0, 0) is situated inside the body. Hi. Background (5) On the basis of Kleinmans( work the investigation of the following problems was proposed: 2(a) Rigorous solution of the integral equation (5), thereby providing a low frequency technique for the scattering problems for acoustically soft objects. 2(b) Derivation of a similar integral equation for the Neumann problem and its rigorous solution. 2(c) Solution ofthwenon-eparable problems, e.g., those problems which are unsolved due to the non-sepaF iy of the Helmholtz equation. 2(d) Extension to vector (electromagnetic) problems. 2(e) The abstract mathematical results which include functional analytic aspects of the problem and the new existence proofs. f 4

THE UNIVERSITY OF MICHIGAN 7359-1-F 2() Extension to two dtmeLn aLo iow -freqeney seartteag problems. 2) Studies in connection with the radius of convergence of the low frequency expansion. Im. The Progress 3(a) The integral equation for the Neumann problem has been found by Ar ^2), and Kleinman With the geometry and the notation indicated in the Introduction, if G is the static Green's function of the second kind and Gk o k the Neumann Green's function for the Helmholtz equation then the integral equation in question is given by ikr o. 1 a uk(p) = -2ike S PP) a,.ikr * / \ AA - + ike Go(p, B) n rB' B r -krl dv1 1 e uk (pl) -tk-B -e- - (pB) do (6) ikr f -e \ B -ikrB akp) Go(PB) e an 3(b) The rigorous solution of this equation (and of that for the Dirichlet case) (4) has been found by Ar(4) This is done by defining a certain function space with a proper norm in which the Neumann series arising from the equations (5) and (6) is convergent to the solution sought. 5

THE UNIVERSITY OF MICHIGAN 7359-1-F 3(c) A surface for which the Helmholtz equation is non-separable is an 2 ogive. However, the Laplace's equation V 0 = 0 is partially separable (andt-m solvable) in the exterior region of this body. Applying the above mentioned techniques the Helmholtz equation has been solved in the "closed" form (for sufficiently small wave numbers) for this case by Ar) M The solution of electromagnetic scattering problems involving a smooth (8) finite three-dimensional scatterer was presented by Stevenson(8) in terms of solutions of standard potential problems involving the same boundary. (6) Kleinman has shown that Stevenson s general procedure leads to erronous field expressions, and in the case when the scatterer is perfectly conducting he has provided a modification which corrects this heretofore-unnoticed deficiency. IV. C9otinuation and the Anticipated New Areas of Investigation 4(a) The function spaces for the Dirichlet as well as the Neumann problems mentioned above, though sufficient to solve these problems, are not complete. (3) However, it has been recently discovered by Ar) that a Banach space can be found which provides new existence (and uniqueness) proofs and rigorous low frequency techniques as well. The preliminary work on this has been completed. 4(b) Another body for which the Helmholtz equation is unsolvable due to its non-separability is the torus. The solution of the torus problem by means similar to that used in '-viFg the gft*e prbtermiftntioned-bovd ':: is being found. These problems are being given the immediate attention at the present, while the other areas mentioned in the original proposal still remain under consideration. 6

THE UNIVERSITY OF MICHIGAN 7359-1-F REFERENCEIS (1) Ar, Ergun (1967), "Low Frequency Scattering from an Ogive' (Submitted to and excepted for publication by Quart. Appl. Math.) (2) Ar, Ergun and R. E. Kleinman, (1967)'The Exterior Neumann Problem for the Three Dimensional Helmholtz Equation", (Submitted to:md excepted for publication by Arch. Rat. Mech. and Anal.) (3) Ar, Ergun, (1966) "On the Solution of Kleinman's Equation in az Banach Space" Notices of Am. Math. Society Vol. 13 No. 7 issue No. 93 (Presented to the joint meeting of Socieded Mathematica Mexicana and the American Mathematical Society November. 25-26, 1966). (4) Ar, Ergun, (1966) "On the Helmholtz Equation for a.Acoisticlly. Rigid Scatterer", University of Michigan Radiation Laboratory Report No. 7359-1-T. (5) Kleinman, R. E. (1965) "The Dirichlet Problem for the Helmholtz Equation", Arch. Rat. Mech. Anal. 1_, No. 3 pp? 205-229. (6) Kleinman, R. E. (1965) "Low Frequency Solution of Electromagnetic Scattering Problems", (Presented at URSI - Delft Symposium, The Netherlands, September 1965). (7) Noble, B. (1962), "Integral Perturbation Methods in Low Frequency Diffraction', Electromagnetic Waves (R. Langer, Ed.). Madison: The University of Wisconsba Press. (8) Stevenson, A. F. (1963) "Solution of Electromagnetic Scattering Problems As Power Series in the Ration (Dimension of Scatterer)/Wavelength", J. Appl. Phys. 24, pp. 1134-1142. 7